$\begingroup$Thanks for the counter-example. Indeed, I am indeed looking at diagonalizable matrices. Can you explain in more detail or sketch out a proof if it is short, or point me to a text that shows this?$\endgroup$
– science404Aug 14 '15 at 21:38

3

$\begingroup$@science404 If a matrix is diagonalizable, it is similar to a diagonal matrix, which is symmetric$\endgroup$
– angryavianAug 14 '15 at 21:38